GENERAL THEORY OF FLEXURE 256. Introduction. The theory of flexure in reinforced concrete is exceptionally complicated. A multitude of simple rules, formula?, and tables for designing reinforced-concrete work have been proposed, some of which are sufficiently accurate and applicable under certain conditions. But the effect of these various conditions should be thoroughly understood. Reinforced concrete should not be designed by "rule-of-thumb" engineers. It is hardly too strong a statement to say that a. man is criminally careless and negligent when he at tempts to design a structure on which the safety and lives of people will depend, without thoroughly understanding the theory on which any formula he may use is based. The applicability of all formuke is so dependent on the quality of the steel and of the concrete, and on many of the details of the design, that a blind application of a formula is very unsafe. Although the greatest pains will be taken to make the following demonstration as clear and plain as possible, it will be necessary to employ symbols, and to work out several algebraic formuke on which the rules for designing will be based. The full significance of many of the terms mentioned below may not be fully understood until several subsequent paragraphs have been studied : b = Breadth of concrete beam; d = Depth from compression face to center of gravity of the steel; A = Area of the steel; p = Ratio of area of steel to area of concrete above the center of gravity of the steel, generally referred to as percentage of re inforcement, A — • b d ' E.= Modulus of elasticity of steel; • E.= Initial modulus of elasticity of concrete; r E'. = Ratio of the moduli; • s = Tensile stress per unit of area in steel; c = Compressive stress per unit of area in concrete at the outer fibre of the beam; this may vary from zero to c'; c' = Ultimate compressive stress per unit of area in concrete—the stress at which failure might be expected; Eo — Deformation per unit of length in the steel; " " " " in outer fibre of concrete; " " " " in outer fibre of concrete when crushing is imminent; c,"= Deformation per unit of length in outer fibre of concrete under a certain condition (described later); q = Ratio of deformations; ,b• k = Ratio of depth from compressive face to the neutral axis to the total effective depth d; x = Distance from compressive face to center of gravity of com pressive stresses; X = Summation of horizontal compressive stresses; = Resisting moment of a section.
257. Statics of Plain Homogeneous Beams. As a preliminary to the theory of the use of reinforced concrete in beams, a very brief discussion will be given of the statics of an ordinary homogeneous beam. Let A B (Fig. SO) represent a beam carrying a uniformly distrib uted load fV; then the beam is subjected to transverse stresses. Let us imagine that one-half of the beam is a "free body" in space, and is acted on by exactly the same external forces; we shall also assume the forces C and T (acting on the exposed section), which are just such forces as are required to keep that half of the beam in equilibrium.
These forces, and their direction, are represented in the lower diagram by arrows. The load !V is represented by the series of small, equal, and equally spaced vertical arrows pointing downward. The reaction of the abutment against the beam is an upward force, shown at the left. The forces acting on a section at the center are the equivalent of the two equal forces C and T.
The force C, acting at the top of the section, must act toward the left, and there is therefore compression in that part of the section. Similarly, the force T is a force acting toward the right, and the fibres of the lower part of the beam arc in tension. For our present purpose we may consider that the forces C and T are in each case the resultant of the forces acting on a very large number of "fibres." The stress in the outer fibres is of course greatest. At the center of the height, there is neither tension nor compression. This is called the neutral axis (see Fig. 90).